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Stress Analysis Concepts

By Vaibhav bhandarkar

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STRESS-STRAIN DIAGRAM

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Most structural materials have an initial region on the stress-strain diagram in which the material behaves elastically and linearly.The linear relationship between stress and strain for a bar in simple tension or compression can be expressed by the equation, commonly known as Hooke’s law, σ = Eε where σ = stress = P/A ε = strain = Δℓ / ℓ E = constant of proportionality known as the Young’s modulus of elasticity ( slope of the stress-strain diagram in the linearly elastic region) = (P x ℓ ) / (A x Δ ℓ )

HOOKE’S LAW

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When a prismatic bar is loaded in tension, the axial elongation is accompanied by lateral contraction (normal to the direction of the applied load).Lateral strain is proportional to the axial strain in the linear elastic region for a homogeneous and isotropic material.The ratio of the strain in lateral direction to the strain in axial direction is known as the Poisson’s ratio which is a dimensionless quantity. ν = (Lateral strain) / (Axial strain) Normally for steels, Poisson’s ratio is 0.3

POISSON’S RATIO

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STIFFNESS

Stiffness is the resistance of an elastic body to deflection or deformation by an applied force.The stiffness of an axially loaded bar is defined as the force required to produce a unit deflection. k = P / Δ ℓ = EA / ℓ

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TYPES OF STRESSES

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Types of Normal stresses – Longitudinal stress

Longitudinal or axial stress is the normal stress acting parallel to the longitudinal axis of the pipe. This may be caused by an internal force acting axially within the pipe. SL = FAX / AM

Where SL = Longitudinal stress FAX = internal axial force acting on cross section Am = metal cross section area of pipe

= π (do2 –di

2) / 4 = π dm t do = Outer dia di = Inner dia dm = Mean dia = (do+ di) / 2

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Longitudinal stress…..cont’d Another example of longitudinal stress is that due to internal pressure. SL = FAX / AM = PAi / AM Where P = Design pressure Ai = internal area of pipe = π di

2 / 4 Hence SL = P(π di

2 /4) / π (do2 –di

2) / 4 = P(di

2 ) / (do2 –di

2) = P(di

2 ) / (do+ di)(do – di) = P(di

2 ) / (2dm)(2t) = P(di

2 ) / (4dmt)For convenience, the longitudinal pressure stress is often approximated as SL= Pdo/ 4t

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Type of Normal stress – Bending stress

Bending stress is zero at the neutral axis of the pipe and varies linearly across the cross section from the maximum compressive outer fibre to the maximum tensile outer fibre.Calculating the stress as linearly proportional to the distance from the neutral axis SL = Mb c / IWhere Mb = bending moment acting at the cross section c = distance of point of interest from the neutral axis I = moment of inertia of the cross section = π (do

4 – di4) /64

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Bending stress…….cont’d

Maximum bending stress occurs where c is greatest – where it is equal to the outer radius Smax = Mb Ro / I = Mb / Z where Ro = outer radius of pipe Z = section modulus of pipe = I / Ro

Summing up all components of longitudinal normal stress :

SL = (Fax / Am) + (Pdo/ 4t) + (Mb / Z)

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Hoop Stress

There are other normal stresses present in the pipe, applied in a directional orthogonal to the axial direction. One of these stresses caused by internal pressure, is called hoop stress. This stress acts in a direction parallel to the pipe circumference.

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Hoop stress ….cont’d

The magnitude of the hoop stress varies through the pipe wall and can be calculated by Lame’s equation as SH = P (ri

2 + ri2 ro

2 / r2 ) / (ro2 - ri

2) where SH = hoop stress due to pressure ri = inner radius of the pipe ro = outer radius of the pipe r = radial position where stress is being considered.The hoop stress can be conservatively approximated for thin walled cylinders, by assuming that the pressure force, applied over an arbitrary length of pipe, ℓ (F=P di ℓ ), is resisted uniformly by the pipe wall over that same arbitrary length (Am = 2 t ℓ ), or SH = P di ℓ / 2 t ℓ = P di / 2 t or conservatively

SH = P do / 2 t

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Radial stresses

Radial stress is the third normal stress present in the pipe wall. It acts in the third orthogonal direction, parallel to the pipe radius.Radial stress which is caused due to internal pressure, varies between a stress equal to the internal pressure at the pipe’s inner surface and a stress equal to the atmospheric pressure at the pipe’s external surface.

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Radial stresses….cont’d

Radial stresses may be calculated as :SR = P (ri

2 - ri2 ro

2 / r2 ) / (ro2 - ri

2) Where SR = radial stress due to pressure.

Note that radial stress is zero at the outer radius of the pipe where the bending stresses are maximized.For this reason, this component has been traditionally been ignored during the stress calculation.

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